{
  "abstract": "We study equality cases in Laffey's lower bound for local maximal commutative subalgebras of matrix algebras with cube-zero radical. We introduce a stacking operation that combines algebras with common outer Loewy layers while preserving maximal commutativity. This yields standard equality bricks $B_{a,c}=(a,ac+1,c;a+c)$ for all $a,c\\ge2$ and reduces further constructions to smaller building blocks. We establish the existence of all minimal equality bricks on the balanced branch and obtain sharp realizations throughout the $(4,5)$-sector. We also construct the first nontrivial nearly balanced root over arbitrary fields. We determine the balanced--nearly-balanced envelope of the numerical Laffey profile and give an arithmetic criterion for preservation of sharpness under stacking. As a consequence, the profile is attained in every matrix dimension $14\\le n\\le52$ and $93\\le n\\le114$, as well as along five infinite families over arbitrary fields. The main existence and interval results are proved in characteristic zero; the field-independent constructions are stated explicitly.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.01387",
  "author_total": 1,
  "authors": [
    {
      "name": "Małgorzata Nowak-Kępczyk"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/45dryrqcafhk8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.01387",
  "fulltext_url": null,
  "id": "45dryrqcafhk8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.01387",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7160278230",
  "pdf_url": "https://arxiv.org/pdf/2605.01387",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/45dryrqcafhk8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2605.01387",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.01387"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Stack Constructions and Equality Cases for Laffey's Bound for Maximal Commutative Matrix Algebras",
  "title_text": "Stack Constructions and Equality Cases for Laffey's Bound for Maximal Commutative Matrix Algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/45dryrqcafhk8",
  "venue": "arXiv",
  "versions": []
}